A theory of valve and circuit noise
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190 ABSTRACTS OF PAPERS the balance point, both when there is no appreciable load across the output terminals and where a resistor R s is connected across the output. In bridges types 1 and 2, the ratio of i?, to R 2 may be altered, but it is shown that unless «, (the exponent of the non-linear element in type 1) or n x ln 2 (the ratio of the exponents of the non- linear elements R 3 and R A in type 2) is much larger or much smaller than unity, there is little to be gained by making this ratio any other than the normal value of unity. The calculations for types 3 and 4 assume a symmetrical arrangement. It is shown that, as regards the figure of merit, types 3 and 4 are superior to types 1 and 2, and that type 4 gives the highest figure of merit. The effect of the value of n and the load (ex- pressed as the ratio R,JR S , where R 4 is the resistance of arm R 4 , which is equal to the resistance of the other arms at balance) is shown in Fig. 2. In the same Figure is shown the value of the figure of merit for type 1 when R t = R 2 . When the bridge is required to operate a relay or indicating instrument, the power output for a given change of input from the balance value is more important than the voltage; formulae are therefore given for the value of R 5 for maximum power out- put. The maximum output for a given change in input may be increased by reducing the resistances of the arms of the bridge, but this increases the power consumption. Some of the more important formulae given in the paper are listed in Table 1. A THEORY OF VALVE AND CIRCUIT NOISE By N. R. CAMPBELL, SC.D., F.Inst.P.,* and V. J. FRANCIS, B.SC, F.Inst.P., Associate Member. [ABSTRACT of a Radio Section paper (communication from the Research Staff of the M.O. Valve Co., Ltd., at the G.E.C. Research Laboratories, Wembley, England) which was published in January, 1946, in Part HI of the Journal.] The conventional theory of the irregularly fluctuating voltages that constitute "noise" relies on a certain equivalence between the irregular fluctuations and sinusoidal oscillations. In this paper an alternative theory is presented, which the authors believe to be physically clearer and which avoids the danger of extending the equivalence beyond the bounds within which it really holds. This theory is a development of ideas much older than the conventional theory. Its basic postulate is that noise consists of the superposed effects of events that are random in time; each event, if it acted alone, would produce an effect y = s(t) depending on the circuit, t being the time since the event began; in linear circuits the superposition is simple addition. Certain very general theorems (Campbell's theorem) give the mean effect and the mean-square effect, resulting from the addition, in terms of s{t) and A, the mean rate at which the events occur: Well-known propositions in operational calculus enable one -to replace s(t) by its Fourier transform, and thus to express the mean effects in terms of the reaction of the circuits to sinusoidal oscillations, i.e. its admittances and gains, as in the conventional iheory. It is shown that this alternative treatment leads very simply to • Formerly a member of the Research Staff of the M.O. Valve Co., Ltd. the accepted formulae for thermal and shot noise in simple cases. More complicated cases are handled by means of the "deter- minant" method of treating valve problems, presented in a recent paper, f which has proved to be a powerful and accurate method of dealing with complicated circuits. Formulae of universal application to linear circuits are given for the thermal mean- square noise and for the shot mean-square noise appearing between any given pair of terminals due to all sources of noise in the circuit. From these formulae, conclusions can be deduced about circuits of any degree of complexity by mere substitution of numerical values for algebraic variables; it is unnecessary to ignore any elements of the circuit or to use intuition to replace an actual circuit by a simpler equivalent. The paper deals primarily with mean-square noise in linear circuits. Its fundamental concepts may, however, be applied to a wider field, although there are probably some problems of non-linear circuits which require a different method of approach. One non-linear problem which can be treated successfully by the theory is the important one of space charge. Here the effects of the events in the valve itself are not additive. Making the same assumptions that North makes, however, it is shown that his results concerning the space-charge reduction of noise follow naturally from the theory. It is formally easy to take into account the finite transit time of the electrons between the electrodes; but the application of the results naturally does not avoid the analytical difficulties of dealing with transit-time effects on sinusoidal oscillations. t CAMPBELL, N. R., FRANCIS, V. J., and JAMES, E. G., "Linear Single-Stage Valve Circuits," Wireless Engineer, 1945, 22, p. 333.
Transcript of A theory of valve and circuit noise
190 ABSTRACTS OF PAPERS
the balance point, both when there is no appreciable load acrossthe output terminals and where a resistor Rs is connected acrossthe output.
In bridges types 1 and 2, the ratio of i?, to R2 may be altered,but it is shown that unless «, (the exponent of the non-linearelement in type 1) or nxln2 (the ratio of the exponents of the non-linear elements R3 and RA in type 2) is much larger or muchsmaller than unity, there is little to be gained by making thisratio any other than the normal value of unity. The calculationsfor types 3 and 4 assume a symmetrical arrangement.
It is shown that, as regards the figure of merit, types 3 and 4are superior to types 1 and 2, and that type 4 gives the highestfigure of merit. The effect of the value of n and the load (ex-
pressed as the ratio R,JRS, where R4 is the resistance of arm R4,which is equal to the resistance of the other arms at balance) isshown in Fig. 2. In the same Figure is shown the value of thefigure of merit for type 1 when Rt = R2.
When the bridge is required to operate a relay or indicatinginstrument, the power output for a given change of input fromthe balance value is more important than the voltage; formulaeare therefore given for the value of R5 for maximum power out-put. The maximum output for a given change in input may beincreased by reducing the resistances of the arms of the bridge,but this increases the power consumption.
Some of the more important formulae given in the paper arelisted in Table 1.
A THEORY OF VALVE AND CIRCUIT NOISE
By N. R. CAMPBELL, SC.D., F.Inst.P.,* and V. J. FRANCIS, B.SC, F.Inst.P., Associate Member.
[ABSTRACT of a Radio Section paper (communication from the Research Staff of the M.O. Valve Co., Ltd., at the G.E.C. Research Laboratories,Wembley, England) which was published in January, 1946, in Part HI of the Journal.]
The conventional theory of the irregularly fluctuating voltagesthat constitute "noise" relies on a certain equivalence betweenthe irregular fluctuations and sinusoidal oscillations. In thispaper an alternative theory is presented, which the authorsbelieve to be physically clearer and which avoids the danger ofextending the equivalence beyond the bounds within which itreally holds.
This theory is a development of ideas much older than theconventional theory. Its basic postulate is that noise consistsof the superposed effects of events that are random in time;each event, if it acted alone, would produce an effect y = s(t)depending on the circuit, t being the time since the event began;in linear circuits the superposition is simple addition. Certainvery general theorems (Campbell's theorem) give the meaneffect and the mean-square effect, resulting from the addition,in terms of s{t) and A, the mean rate at which the events occur:
Well-known propositions in operational calculus enable one-to replace s(t) by its Fourier transform, and thus to express themean effects in terms of the reaction of the circuits to sinusoidaloscillations, i.e. its admittances and gains, as in the conventionaliheory.
It is shown that this alternative treatment leads very simply to• Formerly a member of the Research Staff of the M.O. Valve Co., Ltd.
the accepted formulae for thermal and shot noise in simplecases.
More complicated cases are handled by means of the "deter-minant" method of treating valve problems, presented in a recentpaper, f which has proved to be a powerful and accurate methodof dealing with complicated circuits. Formulae of universalapplication to linear circuits are given for the thermal mean-square noise and for the shot mean-square noise appearingbetween any given pair of terminals due to all sources of noisein the circuit. From these formulae, conclusions can be deducedabout circuits of any degree of complexity by mere substitutionof numerical values for algebraic variables; it is unnecessary toignore any elements of the circuit or to use intuition to replacean actual circuit by a simpler equivalent.
The paper deals primarily with mean-square noise in linearcircuits. Its fundamental concepts may, however, be applied toa wider field, although there are probably some problems ofnon-linear circuits which require a different method of approach.
One non-linear problem which can be treated successfully bythe theory is the important one of space charge. Here the effectsof the events in the valve itself are not additive. Making thesame assumptions that North makes, however, it is shown thathis results concerning the space-charge reduction of noise follownaturally from the theory.
It is formally easy to take into account the finite transit timeof the electrons between the electrodes; but the application ofthe results naturally does not avoid the analytical difficulties ofdealing with transit-time effects on sinusoidal oscillations.
t CAMPBELL, N. R., FRANCIS, V. J., and JAMES, E. G., "Linear Single-Stage ValveCircuits," Wireless Engineer, 1945, 22, p. 333.
